Theorem wpthswwlks2on 29864

Description: For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.)

Assertion

Ref	Expression
wpthswwlks2on	⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Proof of Theorem wpthswwlks2on

Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlknon 29760	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵))
2	1	a1i 11	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
3	2	anbi1d 631	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
4		3anass 1094	. . . . . . 7 ⊢ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
5	4	anbi1i 624	. . . . . 6 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
6		anass 468	. . . . . 6 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
7	5, 6	bitri 275	. . . . 5 ⊢ (((𝑤 ∈ (2 WWalksN 𝐺) ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
8	3, 7	bitrdi 287	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ (𝑤 ∈ (2 WWalksN 𝐺) ∧ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))))
9	8	rabbidva2 3404	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)})
10		usgrupgr 29088	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ UPGraph)
11		wlklnwwlknupgr 29789	. . . . . . . . . . 11 ⊢ (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
12	10, 11	syl 17	. . . . . . . . . 10 ⊢ (𝐺 ∈ USGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) ↔ 𝑤 ∈ (2 WWalksN 𝐺)))
13	12	bicomd 223	. . . . . . . . 9 ⊢ (𝐺 ∈ USGraph → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) ↔ ∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)))
15		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(Walks‘𝐺)𝑤)
16		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘0) = 𝐴)
17	16	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘0) = 𝐴)
18		fveq2 6840	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑓) = 2 → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
19	18	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = (𝑤‘2))
20		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (𝑤‘2) = 𝐵)
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘2) = 𝐵)
22	19, 21	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑤‘(♯‘𝑓)) = 𝐵)
23		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24	23	wlkp 29520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(Walks‘𝐺)𝑤 → 𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺))
25		oveq2 7377	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑓) = 2 → (0...(♯‘𝑓)) = (0...2))
26	25	feq2d 6654	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑓) = 2 → (𝑤:(0...(♯‘𝑓))⟶(Vtx‘𝐺) ↔ 𝑤:(0...2)⟶(Vtx‘𝐺)))
27	24, 26	syl5ibcom 245	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(Walks‘𝐺)𝑤 → ((♯‘𝑓) = 2 → 𝑤:(0...2)⟶(Vtx‘𝐺)))
28	27	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑤:(0...2)⟶(Vtx‘𝐺))
29		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 𝑤:(0...2)⟶(Vtx‘𝐺))
30		2nn0 12435	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℕ0
31		0elfz 13561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (2 ∈ ℕ0 → 0 ∈ (0...2))
32	30, 31	mp1i 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 0 ∈ (0...2))
33	29, 32	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘0) ∈ (Vtx‘𝐺))
34		nn0fz0 13562	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2 ∈ ℕ0 ↔ 2 ∈ (0...2))
35	30, 34	mpbi 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ (0...2)
36	35	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → 2 ∈ (0...2))
37	29, 36	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → (𝑤‘2) ∈ (Vtx‘𝐺))
38	33, 37	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤:(0...2)⟶(Vtx‘𝐺) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
39	28, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)))
40		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤‘0) = 𝐴 → ((𝑤‘0) ∈ (Vtx‘𝐺) ↔ 𝐴 ∈ (Vtx‘𝐺)))
41		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤‘2) = 𝐵 → ((𝑤‘2) ∈ (Vtx‘𝐺) ↔ 𝐵 ∈ (Vtx‘𝐺)))
42	40, 41	bi2anan9 638	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (((𝑤‘0) ∈ (Vtx‘𝐺) ∧ (𝑤‘2) ∈ (Vtx‘𝐺)) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
43	39, 42	imbitrid 244	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
44	43	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
45	44	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
46		vex 3448	. . . . . . . . . . . . . . . 16 ⊢ 𝑓 ∈ V
47		vex 3448	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
48	46, 47	pm3.2i 470	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ V ∧ 𝑤 ∈ V)
49	23	iswlkon 29559	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝑓 ∈ V ∧ 𝑤 ∈ V)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
50	45, 48, 49	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ (𝑓(Walks‘𝐺)𝑤 ∧ (𝑤‘0) = 𝐴 ∧ (𝑤‘(♯‘𝑓)) = 𝐵)))
51	15, 17, 22, 50	mpbir3and 1343	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤)
52		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐺 ∈ USGraph)
53		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (♯‘𝑓) = 2)
54		simpllr 775	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝐴 ≠ 𝐵)
55		usgr2wlkspth 29662	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝑓) = 2 ∧ 𝐴 ≠ 𝐵) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
56	52, 53, 54, 55	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑤 ↔ 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
57	51, 56	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) ∧ (𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2)) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)
58	57	ex 412	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → ((𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
59	58	eximdv 1917	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
60	59	ex 412	. . . . . . . . 9 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
61	60	com23 86	. . . . . . . 8 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (∃𝑓(𝑓(Walks‘𝐺)𝑤 ∧ (♯‘𝑓) = 2) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
62	14, 61	sylbid 240	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝑤 ∈ (2 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
63	62	imp 406	. . . . . 6 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) → ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))
64	63	pm4.71d 561	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ↔ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)))
65	64	bicomd 223	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) ∧ 𝑤 ∈ (2 WWalksN 𝐺)) → ((((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)))
66	65	rabbidva 3409	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (2 WWalksN 𝐺) ∣ (((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
67	9, 66	eqtrd 2764	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤} = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)})
68	23	iswspthsnon 29759	. 2 ⊢ (𝐴(2 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
69	23	iswwlksnon 29756	. 2 ⊢ (𝐴(2 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (2 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘2) = 𝐵)}
70	67, 68, 69	3eqtr4g 2789	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  {crab 3402  Vcvv 3444   class class class wbr 5102  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  0cc0 11044  2c2 12217  ℕ₀cn0 12418  ...cfz 13444  ♯chash 14271  Vtxcvtx 28899  UPGraphcupgr 28983  USGraphcusgr 29052  Walkscwlks 29500  WalksOncwlkson 29501  SPathsOncspthson 29616   WWalksN cwwlksn 29729   WWalksNOn cwwlksnon 29730   WSPathsNOn cwwspthsnon 29732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-ac2 10392  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-map 8778  df-pm 8779  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9830  df-card 9868  df-ac 10045  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-xnn0 12492  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-hash 14272  df-word 14455  df-concat 14512  df-s1 14537  df-s2 14790  df-s3 14791  df-edg 28951  df-uhgr 28961  df-upgr 28985  df-umgr 28986  df-uspgr 29053  df-usgr 29054  df-wlks 29503  df-wlkson 29504  df-trls 29594  df-trlson 29595  df-pths 29617  df-spths 29618  df-pthson 29619  df-spthson 29620  df-wwlks 29733  df-wwlksn 29734  df-wwlksnon 29735  df-wspthsnon 29737

This theorem is referenced by:  usgr2wspthons3  29867  frgr2wsp1  30232